\chapter{{The Model Structure on $\SSets$}}\label{SS_ModelStructure}

http://ncatlab.org/nlab/show/model+structure+on+simplicial+sets

\noindent In this section we recall the canonical Quillen's model structure on the category of simplicial sets, where we will follow \cite{Goe09,Hov07,Pal13}, \tcr{ Andy..}. \tcb{Having a model structure on the category of simplicial sets enables us to study homotopy theory on this category.}

\noindent We will recall the three classes of morphisms of simplicial sets Kan
fibrations, weak equivalence and cofibrations, and study some of its properties.
These classes will form the canonical model structure on the category of
simplicial sets, Th. \ref{Th_SimpSets_ModelStructure}.

\noindent We distinguish two classes of canonical injection of simplicial sets, that play a central role in defining the desired classes of fibrations and cofibrations of the canonical model structure on the category of simplicial sets. Hereby, we fix the notation in this section for:
$$
\begin{array}{l}
\II:=\{\partial_{[n]}:\partial\Delta^n\hookrightarrow \Delta^n|n\in \ZZ_{\geq 0}\}\\
\Lambda:=\{\lambda_r^n:\Lambda_r^n\hookrightarrow \Delta^n|n\in \ZZ_{> 0},0\leq r\leq n\}
\end{array}.
$$
These classes will be the building blocks for (weak) cofibrations and fibrations using \tcb{class operations} $\!-\!$inj and $\!-\!$inj. %Therefore, we start be recalling the some of the criteria of belonging to these classes:
%Motivation





\section{Weak Equivalences}
\begin{definition}[Weak Equivalence]
\textup{Let $f:X\rightarrow Y$ be a morphism of simplicial sets. Then, $f$ is called a weak equivalence if the realisation $\mid f\mid:\mid X\mid\rightarrow \mid Y\mid$ is a weak equivalence of topological spaces, i.e.
\[
\pi_{0}(\mid f\mid):\pi_{0}(\mid X\mid)\rightarrow \pi_{0}(\mid Y\mid)
\]
is a bijection of sets, and for every $n\in \ZZ_{\geq 1}$
\[
\pi_{n}(\mid f\mid,x):\pi_{n}(\mid X\mid,x)\rightarrow \pi_{n}(\mid Y\mid,\abs{f}(x))
\]
is a a group isomorphism for $x\in \mid X\mid$.}
\end{definition}
\Rem{}{Notice that the condition on $\pi_0$ could be formulated using the topological homotopy pointed sets (following definition \ref{Def:PSSets:HomGrp:PSets}), which makes the statement of the condition more consistent, but more redundant. So, it is a matter of taste which condition to choose on $\pi_0$. We prefer to use the consistent notations in formulating statements, and, of course, use the more efficient condition in proofs.
}{}
\Rem{}{Notice that by the construction of the geometric realisation, every path connected component of $\abs{X}$ contains at least a vertex of $X$. Since we have an isomorphism
\[
\pi_{n}(\mid X\mid,x)\cong \pi_{n}(\mid X\mid,x')
\]
for every pair of points $x,x'\in X$ that lies in the same path connected component, then the above definition can be given more efficiently by restricting $x$ to the vertices of $X$. Actuality, it is enough to restrict the choice of $x$ to one point in each path connected components. However, in practice, computing the path connected components of a given simplicial sets may require much more effort than considering all the vertices.}{}

Weak equivalence can be defined equivalently using the base-free homotopy sets, as in the below lemma

\Lem{}{Let $f:X\rightarrow Y$ be a morphism of simplicial sets. Then, the following statements are equivalent
\begin{itemize}
\item $f$ is a weak equivalence.
\item $\forall n\geq 0$, the induced square, constructed in \eqref{eq:SSets:BFHomGrp:Square}, is a cartesian square.
\[
\xymatrix{
\pi^{\Top}_n(X)\ar[d]_{p_X}\ar[r]^{f_\ast}\cd&\pi^{\Top}_n(Y)\ar[d]^{p_Y}\\
X_0\ar[r]_{f_0}&Y_0
}
\]
\end{itemize}
}{}
\begin{proof}
A direct result of the above remark and corollary \ref{Cor:SSets:PullBack:Isomorphism}.
\end{proof}
\tcb{type examples of weak equivalences}




\section{Cofibrations}
\begin{definition}[Cofibrations of $\SSets$]\label{SSets:Cof:Def}
A morphism of simplicial sets $i:X\rightarrow Y$ is said to be a cofibration if
it is in $\II\!-\!$cof.
\end{definition}


\begin{lemma}\label{SSets:Cof:Mono}
A morphisms $i:X\rightarrow Y$ of simplicial sets is a cofibrations if and
only if it is an inclusion (a monomorphism of simplicial sets). Hence, all simplicial sets are cofibrant.
\end{lemma}
\begin{proof}

\tcb{Joe 2.4.4}


\end{proof}
\begin{lemma}
Cofibrations of simplical sets are relative $\II\!-\!$cell complexe, i.e.
$\II\!-\!$cof$\subset \II\!-\!$cell - that the other inclusion is always
satisfied for any class of morphisms, by
\ref{Lem:InjProjCellClosed}(\ref{itm:CellCof}).
\end{lemma}

\begin{proof}
\tcr{Joe 2.4.4}
\end{proof}


\subsection{Anodyne Extension}
\begin{definition}[Anodyne Extension]
A morphism of simplicial sets $i:X\rightarrow Y$ is said to be a anodyne extension if
it is in $\JJ\!-\!$cof.
\end{definition}

\begin{lemma}
A morphism of simplicial sets is a weak cofibrations iff it is an anodyne extensions. 
\end{lemma}
\begin{proof}
\tcr{2.4.5}
\end{proof}
\begin{remark}
Anodyne extensions are cofibrations, but not necessarily weak equivalences.
\end{remark}

tcr{Moore}



\subsubsection{Product and Anodyne Extensions}

\begin{proposition}\label{Prop:ProdAno} Let $i:K\rightarrow L$ be an inclusion of simplicial sets, and $j:U\rightarrow V$ be an anodyne extension. Then, $j\Box i$ is an anodyne extension.
\end{proposition}
\begin{proof}
\tcb{Joe 2.4.19, b is a result of (a), the fact that $\JJ\!-\!$inj=$(\JJ\!-\!$cof$)\!-\!$inj and the previous lemma.}
\end{proof}




\begin{corollary}Let $i:K\rightarrow L$ be an inclusion of simplicial sets, $j:U\rightarrow V$ be an Anodyne extension. Then:
\begin{enumerate}
\item $j\times id_L:U\times L\rightarrow V\times L$ is an anodyne extension.
\item $id_V\times i:V\times K\rightarrow V\times L$ is an anodyne extension.
\end{enumerate}
\end{corollary}
\begin{proof}The unique morphisms $i':\emptyset\rightarrow L$, and $j':\emptyset\rightarrow V$ are inclusions of simplicial sets.\\ 
Notice that $U\times\emptyset=V\times\emptyset=\emptyset \times K=\emptyset \times L =\emptyset$. Hence,
\begin{enumerate}
\item $j\coprod i'=\emptyset\coprod_\emptyset (U\times L)=U\times L$. Thus, $j\times id_L$ coincide with $j\Box i'$, and by proposition \ref{Prop:ProdAno}, $j\times id_L$ is an anodyne extension.
\item $j'\coprod i=(V\times K)\coprod_\emptyset \emptyset=V\times K$. Thus, $id_V\times i$ coincide with $j'\Box i$, and by proposition \ref{Prop:ProdAno}, $id_V\times i$ is an anodyne extension.
\end{enumerate}
\end{proof}





\section{Kan Fibrations}
\begin{definition}[Kan Fibrations]
\textup{A morphism of simplicial sets $p:X\rightarrow Y$ is said to be a fibration (Kan fibration) if it has the RLP with respect to all canonical injections $\Lambda_r^n\hookrightarrow \Delta^n$ for $n\in \ZZ_{> 0},0\leq r\leq n$, i.e. the class of Kan fibrations is $\JJ\!-\!$inj. Moreover, we call a simplicial set $X$ Kan fibrant (Kan complex) if the unique morphism $X\rightarrow \ast$ is a Kan fibration.}
\end{definition}

\noindent the condition on a simplicial set $X$ to be a Kan fibrant used to be called the \emph{extension condition}, because it means that every morphism of simplicial sets $\Lambda_r^n\rightarrow X$ can be extended to  morphism $h:\Delta^n\rightarrow X$, that the below solid diagram is always commutative.
$$\
\xymatrix{
\Lambda_r^n\ar[r]\ar@{^(->}[d]_{\lambda_r^n}&X\ar[d]\\
\Delta^n\ar[r]\ar@{..>}[ur]|h&\ast
}
$$

The extension condition is used to be given by combinatorial relations, we recall the equivalence between the two definitions in the below lemma:
\begin{lemma}[Extension condition]
Let $p:X\rightarrow Y$ be a morphism of simplicial sets, then the below two statements are equivalents:
\begin{enumerate}
\item\label{itm:KanEx1} $p$ is a Kan fibration.
\item\label{itm:KanEx2} $\forall n\in \ZZ_{> 0}, 0\leq r \leq n$:\\
For every $n\!-\!$tuple of $(n-1)\!-\!$simplices $(x^0_{n-1},x^1_{n-1},...,\widehat{x^r_{n-1}},...,x^n_{n-1})$ of $X$ such that $d_i^{n-1}(x^j_{n-1})=d_{j-1}^{n-1}(x^i_{n-1})$ for $r\neq i<j\neq r$ , and for every $n\!-\!$simplex $y_n\in Y_n$ such that $d_i^n(y_n)=p(x^i_{n-1}), i\neq r$, there exists $x_n\in X_n$ such that $d_i^n(x_n)=x^i_{n-1}$, and $p_n(x_n)=y_n$. Where $\widehat{}$ indicates a deleted entry.
\end{enumerate}
\end{lemma}
\begin{proof}
Since $\Lambda_r^n$ is generated by, $n$ many of $(n-1)\!-\!$simplex, $\partial^i_{n}$, $0\leq i\neq r\leq n$, that satisfy the set minimal simplicial relations \eqref{Eq:G&R:Horn}, then by lemma \ref{Simp:Sets:MorphismsByGenerators} we see that giving a morphism $\Lambda_r^n\rightarrow X$ is equivalent to giving $n\!-\!$tuple of $(n-1)\!-\!$simplices $(x^0_{n-1},x^1_{n-1},...,\widehat{x^r_{n-1}},...,x^n_{n-1})$ of $X$ such that $d_i^{n-1}(x^j_{n-1})=d^{n-1}_{j-1}(x^i_{n-1})$ for $r\neq i<j\neq r$. Also, since $\Delta^n$ is generated by the $n\!-\!$simplex $id_{[n]}$ and the empty set of minimal simplicial relations, then giving a morphism $\Delta^n\rightarrow Y$ is equivalent to giving an $n\!-\!$simplex $y_n\in Y_n$. In the rest of the proof, $0\leq i\neq r \leq n$, and $0\leq j \neq r \leq n$:
\begin{itemize}
\item[{\ref{itm:KanEx1}$\Rightarrow$\ref{itm:KanEx2}}] Assume that $p$ is a Kan fibration, and assume that the hypothesis of (\ref{itm:KanEx2}) holds, then there is a solid diagram
$$
\xymatrix{
\Lambda_r^n\ar[r]^{e_0}\ar@{^(->}[d]_{\lambda_r^n}&X\ar[d]^p\\
\Delta^n\ar[r]_{e_1}\ar@{..>}[ur]|h&Y
}.
$$
where $e_{0,n-1}(\partial_{n}^i)=x_{n-1}^i$, and $e_{1,n}(id_{[n]})=y_n$. By \ref{Simp:Sets:MorphismsByGenerators}, this diagram is commutative iff it is commutative on generators. Since $d_i^n(y_n)=p_{n-1}(x^i_{n-1})$, then $d_i^n(e_{1,n}(id_{[n]}))=p_{n-1}(e_{0,n-1}(\partial_{n}^i))$. $e_1$ is a morphism of simplicial sets, then $d^n_i e_{1,n}=e_{1,n-1}d^{n-1}_i$, hence:
$$p_{n-1}(e_{0,n-1}(\partial_{n}^i))=e_{1,n-1}(d^{n-1}_i(id_{[n]})=e_{1,n-1}(\partial_{n}^i)=e_{1,n-1}(\lambda_{r,n-1}^n(\partial_{n}^i))$$
i.e, $pe_0=e_1 \lambda_r^n$, and the diagram is commutative. Then, there exists a lift $h:\Delta^n\rightarrow X$, let $x_n=h(id_{[n]})$, then the commutativity of the whole diagram implies that $p_n(x_n)=y_n$, and 
$$\begin{array}{ll}
d_i^n(x_n)&=d_i^n(h_n(id_{[n]}))=h_{n-1}(d_i^n (id_{[n]}))=h_{n-1}(\partial_{n}^i)=h_{n-1}(\lambda_{r,n-1}^n(\partial_{n}^i))=e_{0,n-1}(\partial_{n}^i)\\&=x_{n-1}^i.
\end{array}
$$

\item[{\ref{itm:KanEx2}$\Rightarrow$\ref{itm:KanEx1}}] Assume (\ref{itm:KanEx2}), and let $e_0$ and $e_1$ be morphisms of simplicial sets that makes the following solid diagram commutative:
$$
\xymatrix{
\Lambda_r^n\ar[r]^{e_0}\ar@{^(->}[d]_{\lambda_r^n}&X\ar[d]^p\\
\Delta^n\ar[r]_{e_1}\ar@{..>}[ur]|h&Y
}.
$$
The morphism of simplicial sets $e_0$ implies the existence $n\!-\!$tuple of $(n-1)\!-\!$simplices $$(e_{0,n-1}(\partial^0_{n}),e_{0,n-1}(\partial^1_{n}),...,\widehat{e_{0,n-1}(\partial^r_{n})},...,e_{0,n-1}(\partial^n_{n}))$$ of $X$ such that $d_i^{n-1}(e_{0,n-1}(\partial^j_{n}))=d_{j-1}^{n-1}(e_{0,n-1}(\partial^i_{n}))$ for $ i<j$. The morphism of simplicial sets $e_1$ distinguishes the $n\!-\!$simplex $y_n=e_{1,n}(id_{[n]})\in Y_n$. Whereas the commutativity of the solid diagram implies that
$$
\begin{array}{ll}
d_i^n(y_n)&=d_i^n(e_{1,n}(id_{[n]}))=e_{1,n-1}(d_i^n(id_{[n]}))=e_{1,n-1}(\partial_n^i)=e_{1,n-1}(\lambda_{r,n-1}^n(\partial_n^i))=p_{n-1}(e_{0,n-1}(\partial_n^i))\\
&=p_{n-1}(x^i_{n-1}).
\end{array}$$
Hence, there exists $x_n\in X_n$ such that $d_i^n(x_n)=e_{0,n-1}(\partial_n^i)$, and $p_n(x_n)=y_n$.
Then, since $\Delta^n$ is generated by $id_{[n]}$ and the empty set of simplicial relations, we define $h:\Delta^n\rightarrow X$ to be the unique morphism of simplicial sets such that $h_n(id_{[n]})=x_n$. We need to see that $h$ makes the whole diagram commutative.
Since $d_i^n(x_n)=e_{0,n-1}(\partial_n^i)$, then:
$$
e_{0,n-1}(\partial_n^i)=d_i^n(h_n(id_{[n]}))=h_{n-1}(d_i^n(id_{[n]}))=h
_{n-1}(\partial_n^i)=h_{n-1}(\lambda_{r,n-1}^n(\partial_n^i))
$$
i.e. $e_0=h\lambda_r^n$. Since $p_n(x_n)=y_n$, then
$e_{1,n}(id_{[n]})=y_n=p_n(x_n)=p_n(h_n(id_{[n]})),$ i.e. $e_1=ph$, and the whole diagram is commutative.
\end{itemize}\!
\end{proof}

We recall below a classical example of Kan fibrants, that links Kan fibrants to Serre fibrants, but before we need to recall the following topological fact:
\begin{lemma}
Let $n\in \ZZ_{> 0}, 0\leq r\leq n$, then $\Lambda_{r,\Top}^n$ is a retract of $\Delta^n_\Top$.
\end{lemma}
\begin{proof}
Define:
$$
\begin{array}{lllll}
r^r_n:	&\Delta^n_\Top	&\longrightarrow	&\Lambda_{r,\Top}^n	&\\
		&t			&\longmapsto		& u,				&u_i=
	\left\{
	\begin{array}{lr}
	t_i-\displaystyle\min_{0\leq j\neq r \leq n}\{t_i\}	&i\neq r\\\\
	1-\displaystyle\sum_{0\leq j\neq r \leq n} u_i		&i=r.
	\end{array}\right.
\end{array}
$$
\noindent $r^r_n$ is a well-defined continuous map. For $t\in \Lambda_{r,\Top}^n\subset \Delta^n_\Top,\exists 0\leq j\neq r \leq n,$ such that $t_i=0$, i.e. $\displaystyle\min_{0\leq j\neq r \leq n}\{t_i\}=0$, and $r^r_n(t)=t$.
\end{proof}
\begin{example}
Let $X$ be a topological space, then  $S(X)$, the singular simplicial set of $X$, is a Kan fibrant.
\end{example}
\begin{proof}
Let $f_{n-1}^i:\Delta^{n-1}_\Top\rightarrow X$ for ${0\leq j\neq r \leq n}$ be $n$ $(n-1)\!-\!$simplices of $S(X)$, such that 
$$
d^{n-1}_i(f_{n-1}^j)=d^{n-1}_{j-1}(f_{n-1}^i)\text{ for } r\neq i\leq j \neq r.
$$
Define
$$
\begin{array}{llll}
f_n:	&\Delta^n_\Top	&\longrightarrow	&X	\\
		&t			&\longmapsto		& f_{n-1}^i((r^r_n(t))_0,(r^r_n(t))_1,...,\widehat{(r^r_n(t))_i},...,(r^r_n(t))_n)\text{ where } (r^r_n(t))_i=0.
\end{array}
$$
\noindent \tcb{The above conditions on $f_{n-1}^i$'s implies that $f_n$ is well-defined. Also, it is continuous, i.e. $f_n\in S(X)_n$. It is easy to see that the Kan extension condition is satisfied for the choice of $f_n$, and that $S(X)$ is a Kan fibrant.}
\end{proof}

\noindent Kan fibrations are linked to Serre fibration through the following lemma and Quillen's Theorem \ref{Th:Quillen}.
\begin{lemma}
Let $p:X\rightarrow Y$ be a continuous map of topological spaces. Then, $p$ is a Serre fibration iff $S(p):S(X)\rightarrow S(Y)$ is a Kan fibration.
\end{lemma}
\begin{proof}
It is a direct result of corolloray \ref{Adj:Lifting}, for $\bcC=\SSets$,
$\bcD=\Top$, $(L,R,\phi)=(\mid - \mid,S,\varphi)$ the adjundtion between
geometric realisation and the singular functor, $I=\JJ$, and
$J=\{p\}$.
\end{proof}


\tcr{finish the exasmple using Jardine 12, and mention the lemma}.

\tcr{Examples of Kan Fibrations, and counter examples.}
\begin{example}[Counter Example]
Consider the solid commutative square:
\noindent \begin{center}
\begin{minipage}{.14\linewidth}
\noindent 
\begin{tikzpicture}[scale=.7]
	\draw [red, thick] ( -0.8660 ,0.5) -- (0,1) -- (-.5,- 0.8660);
	\node at (.2,1) {$0$};
	\node at (-.4,- 1.1) {$1$};
	\node at ( -0.9660 ,0.7) {$2$};
	\end{tikzpicture}\\
	\xymatrix@C=1.2pc{&\ar@{^(->}[d]\\& \ }\\
	%\begin{tikzpicture}[scale=1]
	%\draw [black, fill=blue,thick] (0,1) -- (-0.577350269,0) -- ( 0.577350269,0)
	% -- (0,1);
	%\draw [red, thick] (-0.577350269,0) -- (0,1) --  ( 0.577350269,0);
	%\node at (0,1.2) {$0$};
	%\node at ( 0.677350269,0) {$1$};
	%\node at ( -0.677350269,0) {$2$};
	%\end{tikzpicture}
	\begin{tikzpicture}[scale=.7]
	%\draw [black,fill=lightgray] (0,0)-- (-.5,- 0.8660) -- (0,1) -- (0,0);
	\draw [black, fill=blue,thick] (0,1) -- (-.5,- 0.8660) -- ( -0.8660 ,0.5) --
	(0,1); \draw [red, thick] ( -0.8660 ,0.5) -- (0,1) -- (-.5,- 0.8660);
	\node at (.2,1) {$0$};
	\node at (-.4,- 1.1) {$1$};
	\node at ( -0.9660 ,0.7) {$2$};
	%\node at (.2,0) {$3$};
	\end{tikzpicture}
\end{minipage}
\noindent \begin{minipage}{.3\linewidth}
\noindent 
\begin{center}
\xymatrix @R=5pc@C=5pc
{
\Lambda_0^2\ar@{^(->}[r]\ar@{^(->}[d]_{\lambda_0^2}&\partial\Delta^2\ar@{^(->}[d]^{\Delta^{\partial_3^3}|_{\partial\Delta^2}}\\
\Delta^2\ar@{^(->}[r]_{\Delta^{\partial_3^3}}\ar@{-->}[ru]|\backslash^h&\Delta^3
}
\end{center}
\end{minipage}
\noindent \begin{minipage}{.14\linewidth}
\noindent 
\begin{tikzpicture}[scale=.7]
%\draw [black, fill=lightgray] (0,0)-- (-.5,- 0.8660) -- (0,1) -- (0,0);
%\draw [black] (-0.232052631,0.133979579)-- (0,1) -- ( -0.8660 ,.5) -- (-0.232052631,0.133979579); 
%\draw [black, fill=gray] (-0.232052631,0.133979579)-- (-.5,- 0.8660) -- (
% -0.8660 ,.5) -- (-0.2598,.15); \draw [black] (0,1) -- (-.5,- 0.8660) -- ( -0.8660 ,0.5) -- (0,1);
\draw [red, thick] ( -0.8660 ,0.5) -- (0,1) -- (-.5,- 0.8660);
\draw [thick] ( -0.8660 ,0.5)  -- (-.5,- 0.8660);
\node at (.2,1) {$0$};
\node at (-.4,- 1.1) {$1$};
\node at ( -0.9660 ,0.7) {$2$};
%\node at (.2,0) {$3$};
\end{tikzpicture}\\ 
\xymatrix@C=.8pc{&\ar@{^(->}[d]\\& \ }\\
\begin{tikzpicture}[scale=.7]
\draw [black,fill=lightgray] (0,0)-- (-.5,- 0.8660) -- (0,1) -- (0,0);
\draw [black, fill=blue,thick] (0,1) -- (-.5,- 0.8660) -- ( -0.8660 ,0.5) --
(0,1); \draw [red, thick] ( -0.8660 ,0.5) -- (0,1) -- (-.5,- 0.8660);
\node at (.2,1) {$0$};
\node at (-.4,- 1.1) {$1$};
\node at ( -0.9660 ,0.7) {$2$};
\node at (.2,0) {$3$};
\end{tikzpicture}
\end{minipage} 
\end{center}

\noindent This diagram does not admit a
lift $h:\Delta^2\rightarrow \partial\Delta^2$, that if we assume for the sake
of contradiction that it does then the commutativity of the whole diagram
implies that 
$$\Delta^{\partial_3^3}_2(id_{[2]})={\Delta^{\partial_3^3}|_{\partial\Delta^2}}_2(h_2(id_{[2]}))$$
i.e. $\partial_3^3=\partial_3^3 h_2(id_{[2]})$ in $\Simp$. Since $\partial_3^3$ is
a monomorphism, in $\Simp$, we have $h_2(id_{[2]})=id_{[2]}\nin
\partial\Delta^2$. We got a contradition, and such $h$ does not exist, i.e.
$\Delta^{\partial_3^3}|_{\partial\Delta^2}$ is not a Kan fibration.
\end{example}
\begin{remark}
Moreover, all the morphisms of $\II$ and $\JJ$ are not Kan fibration,
that we have the commutative squares
$$id_{\lambda_r^n},id_{\partial_{[n]}}\in Mor(\SSets)\forall n\in
\ZZ_{\geq 0}, 0\leq r\leq n.$$
One can see easily that neither of these squares has a lift, that the existance
of such lift requires $id_{[n]}\in \Lambda_r^n, id_{[n]}\in
\partial\Delta^n$, respectively, which is not the case, and elements of
$\II$ and $\JJ$ are not Kan fibration.\\

Also, notice that $\forall n\in \ZZ_{\geq 0}, 0\leq r\leq n, \Delta^n$ is a Kan fibrant, whereas $\Lambda_r^n,$ and $\partial\Delta^n$ are not. \tcr{That}
\end{remark}
\begin{example}
Let $X$ be a simplicial set. Then, we have a unique morphism $\emptyset\rightarrow X$. Since $\Lambda^0_0\hookrightarrow \Delta^0$ is not in $\JJ$, we do do not need to consider whether a lift exist for the commutative square
\[
\xymatrix{
\emptyset=\Lambda_0^0\ar[r]\ar[d]&\emptyset\ar[d]\\
\ast=\Delta^0\ar[r]_{\iota_{x_0}}&X.
}
\]
In fact, such lift does not exist. Actually, the condition of $\emptyset\rightarrow X$ being a Kan fibration is satisfied automatically.

On the other hand, one can see that $\emptyset\rightarrow X$ is Kan fibration as a result of the functorial factorisation as a fibration and weak cofibration and the fact that $\emptyset$ is only weak equivelant to itself.
\end{example}


\begin{remark}\label{KanPull}
Notice that the fibres of Kan fibration are Kan fibrant that, by Lemma \ref{Lem:InjProjCellClosed}\ref{itm:InjPullProjPush}.
\end{remark}



\noindent Since the aim of this section is to recall the canonical model structure on the category of simplicial sets. It is of a particular interest to study Kan fibrations, which are also weak equivalences, called weak fibrations. However, the study of weak fibrations is not easy, and the method we adopt here requires familiarity with function complexes, simplicial homotopy and minimal fibrations. Therefore, we devote the rest of this section to recall these concept and eventually classify weak fibrations.





\subsection{Function Complexes and Kan Fibrations}



\begin{proposition}\label{Prop:HomKan} Let $i:K\rightarrow L$ be an inclusion of simplicial sets, and $p:X\rightarrow Y$ be a Kan fibration. Then, $(i^\ast,p_\ast)$ is a Kan fibration.
\end{proposition}
\begin{proof}
\tcb{Joe 2.4.19, b is a result of (a), the fact that $\JJ\!-\!$inj=$(\JJ\!-\!$cof$)\!-\!$inj and the previous lemma.}
\end{proof}
\begin{corollary}\textup{Let $i:K\rightarrow L$ be an inclusion of simplicial sets, $p:X\rightarrow Y$ be a Kan fibration, and $X'$ be a Kan fibrant. Then:
\begin{enumerate}
\item $p_\ast:\Hom(K,X)\rightarrow \Hom(K,Y)$ is a Kan fibration.
\item $i^\ast:\Hom(L,X')\rightarrow \Hom(K,X')$ is a Kan fibration.
\end{enumerate}}
\end{corollary}
\begin{proof}The unique morphism $p':X'\rightarrow \ast$ is a Kan fibration.\\ 
Notice that $\Hom(\emptyset,X)=\Hom(\emptyset,Y)=\Hom(L,\ast)=\Hom(K,\ast)=\ast$.
\begin{enumerate}
\item  $i'^\ast\prod p_\ast=\ast\times_{\ast}\Hom(L,Y)=\Hom(L,Y)$. Thus, $p_\ast$ coincide with $(i'^\ast,p^\ast)$, and by proposition \ref{Prop:HomKan}, $p_\ast$ is a Kan fibration.
\item $i^\ast\prod p'_\ast=\Hom(K,X)\times_\ast \ast=\Hom(K,X)$. Thus, $i^\ast$ coincide with $(i^\ast,p'^\ast)$, and by proposition \ref{Prop:HomKan}, $i^\ast$ is a Kan fibration.
\end{enumerate}
\end{proof}
















\subsection{Homotopy of Simplicial Sets}

We saw before that simplicial homotopy is not an equivalence relation on the class of morphisms of simplicial sets. However, the below lemma provides the framework where the simplicial homotopy is an equivalence relation.

\subsection{Simplicial Homotopy Groups} 
\noindent Below, we recall the simplicial homotopy groups that are constructed in an analogue way of constructing the topological homotopy groups. Where, first we distinguish morphisms of simplicial sets from the standard simplexes to a simplicial sets $X$, that are constant on the boundary. Then, we define then the simplicial homotopy groups of $X$ to be the equivalences classes of the simplicial homotopy of such morphisms. However, as we have seen before, simplicial homotopy is not an equivalence relation in general. Therefore, the definition of the simplicial homotopy groups will be restricted to the case were $X$ is a Kan fibrant, that the below lemma  guarantees that the simplicial homotopy of such morphisms forms an equivalence relation. Then, we recall the relation that one is expecting between the simplicial homotopy groups of $X$, and the the homotopy groups of its geometric realisation.\\
\begin{lemma}
Let $X$ be a Kan fibrant, and $i:K\rightarrow L$ be an inclusion of simplicial sets. Then,
\begin{enumerate}
\item The simplicial homotopy of morphisms $L\rightarrow X$ is an equivalence relation.
\item The simplicial homotopy of morphisms $L\rightarrow X$ (rel $K$) is an equivalence relation.
\end{enumerate}
Moreover, the graded equivalence relation defined on $X$, by identifying $n\1-\!$simplices of $X$ with morphisms $\Delta^n\rightarrow X$, respects faces and degenerations.
\end{lemma}
\begin{proof}
\tcr{type}.
\end{proof}

\begin{lemma}
Let $p:X\rightarrow Y$ be a Kan fibration, and $i:K\rightarrow L$ be an inclusion of simplicial sets. Then,
\begin{enumerate}
\item The simplicial homotopy of morphisms $L\rightarrow X$ with respect to $p$ is an equivalence relation.
\item The simplicial homotopy of morphisms $L\rightarrow X$ (rel $K$) with respect to $p$ is an equivalence relation.
\end{enumerate}
Moreover, the graded equivalence relation defined on $X$, by identifying $n\1-\!$simplices of $X$ with morphisms $\Delta^n\rightarrow X$, respects faces and degenerations.
\end{lemma}
\begin{proof}
\tcr{Type}.
\end{proof}



\noindent In topological homotopy groups we consider homotopy classes at fixed basepoints, below we recall how to make sense of these concepts for simplicial homotopy.
\begin{definition}[\tcr{Loops at vertex}]
Let $X$ be a simplicial set, $x_0\in X_0$ be a vertex of $X$. Let $x_n\in X_n$, we say that $x_n$ is a loop at $x_0$ if its boundary being pure degeneracy of $x_0$, i.e. if $\iota_{x_n}$ being constant to $x_0$ on the boundary $\partial\Delta^n$, which is equivalent of requiring the below diagram to commute:
$$
\xymatrix{
\partial\Delta^n\ar@{^(->}[d]_{\partial_{\Delta^n}}\ar[r]&\Delta^0=\ast\ar[d]^{\iota_{x_0}}\\
\Delta^n\ar[r]_{\iota_{x_n}}&X
}
$$
We use the same notation to say that $\iota_{x_n}$ is a loop at $x_0$.
\end{definition}
\begin{lemma}\label{Lem:LoopBoundary}
Let $X$ be a simplicial set, $x_0\in X_0$, and $x_n\in X_n$. Then, $x_n$ is a loop at $x_0$ iff $\forall 0\leq i\leq n$, $d_i^n(x_n)=\iota_{x_0,n-1}(\sigma_{n-1})$. Hence, $\partial(x_n)=(\iota_{x_0,n-1}(\sigma_{n-1}),\iota_{x_0,n-1}(\sigma_{n-1}),...,\iota_{x_0,n-1}(\sigma_{n-1}))$.
\end{lemma}
\begin{proof}
The commutativity of the below diagram:
$$
\xymatrix{
\Delta^n_n\ar[d]_{\Delta^n_{\partial_n^i}}\ar[r]^{\iota_{x_0,n}}&X_n\ar[d]^{d_i^n}\\
\Delta^n_{n-1}\ar[r]_{\iota_{x_0,n-1}}&X_{n-1}
}
$$
implies that
$$
d_i^n(x_n)=\iota_{x_n,n-1}(\partial_n^i)
$$
Since $\partial\Delta^n$ is generated by $\partial_n^i$'s, then $d_i^n(x_n)\in \im(\xymatrix{\partial\Delta^n_{n-1}\ar@{^(->}[r]&\Delta^n\ar[r]^{\iota_{x_n}}&X})$.\\

\noindent On the one hand, if $x_n$ is a loop at $x_0$, then $\im(\xymatrix{\partial\Delta^n_{n-1}\ar@{^(->}[r]&\Delta^n\ar[r]^{\iota_{x_n}}&X})=\{X_{\sigma_{n-1}}(x_0)=\iota_{x_0,n-1}(\sigma_{n-1})\}$, hence $d_i^n(x_n)=\iota_{x_0,n-1}(\sigma_{n-1})$.\\

\noindent On the other hand, having $d_i^n(x_n)=\iota_{x_0,n-1}(\sigma_{n-1})$, $\forall 0\leq i\leq n$ implies that $\xymatrix{\partial\Delta^n_{n-1}\ar@{^(->}[r]&\Delta^n\ar[r]^{\iota_{x_n}}&X}$ equals the constant morphisms on $\partial\Delta^n$ to $x_0$, on generators, hence they coincide, and $x_n$ is a loop at $x_0$.
\end{proof}


\begin{definition}
\textup{Let $X$ be a Kan fibrant, $x_0\in X_0$ be a vertex of $X$. Then, for $n\in \ZZ_{\geq 0}$ $\pi_n(X,x_0)$ is defined to be the set of homotopy classes of loops $\Delta^n\rightarrow X$ at $x_0$ relative to the boundary the  $\partial\Delta^n$, i.e. the set of homotopy classes of morphisms $f:\Delta^n\rightarrow X$ (rel $\partial\Delta^n$) for which the below diagram commutes:
$$
\xymatrix{
\Delta^n\ar[r]^f&X\\
\partial\Delta^n\ar@{^(->}[u]^{\partial_{\Delta^n}}\ar[r]&\Delta^0=\ast\ar[u]_{\iota_{x_0}}
}
$$
Moreover, $X$ is said to be connected if $\pi_0(X)$ consists of one element, the change of notation of dropping the vertex from $\pi_0$ is explain in remark \ref{Rem:Pi_0}, below.}
\end{definition}
\begin{remark}
Since $f$ is a loop at $x_0$, then the requirement of the simplicial homotopy in the above definition to be relative to $\partial\Delta^n$ is to say that all the element of the desired equivalent class are also loops at $x_0$. If we think in terms of geometric realisation, it is the analogue of finding a homotopy classes of loops going through some fixed point of the topological space, in order to construct the topological fundamental group.\\
\end{remark}
\begin{remark}\label{Rem:Pi_0}
Notice that for $n=0$, $\partial\Delta^0=\emptyset$, hence any morphism $f:\Delta^0\rightarrow X$ satisfy the above condition on $f$ automatically, and $\pi_0(X,x_0)$ is then the set of homotopy classes of vertices of $X$, and it is independent of the choice of $x_0$, therefore we denote it by $\pi_0(X)$.
\end{remark}
\begin{lemma}
Let $X$ be a Kan fibration, $x_0\sim x'_0\in X_0$, then $\forall n\in \ZZ_{\geq 0}$, there is a bijection $\pi_n(X,x_0)\cong \pi_n(X,x'_0)$.
\end{lemma}
\begin{proof}
\tcb{type}
\end{proof}
\noindent Hence, if $X$ is connected then we can drop the vertex of the notation, and write $\pi(X)$.

\noindent in an analogue of the topological homotopy groups, one would like to equip $\pi_n(X,x_0)$ with a group structure. \tcr{(This needs to be changed)}
\begin{definition}\label{Def:P_nOperation}
Let $X$ be a Kan fibration, $x_0\in X_0$, and $ x_n,x'_n\in X_n$ are loops at $x_0$. Then, consider the $(n+1)\!-\!$tuple $(x_n^0,x_n^1,...,x_n^n,\widehat{x_n^{n+1}})$ of $n\!-\!$simplices of $X$, where:
$$
x_n^i=
\left\lbrace
\begin{array}{lr}
\iota_{x_0,n}(\sigma_n)&0\leq i\leq n-2\\
x_n&i=n-1\\
x'_n&i=n
\end{array}\right.
$$
Notice that lemma \ref{Lem:LoopBoundary} and remark \ref{Rem:PureDegVer} imply that $d_i^n(x_n^j)=\iota_{x_0,n-1}(\sigma_{n-1})$ for $0\leq i\leq n$ and $0\leq j \leq n$. Then in particular $d_i^n(x_n^j)=d_{j-1}^n(x_n^i)$ for $n+1 \neq i<j\neq n+1$. Since, $X$ is a Kan fibrant, then there exists an $(n+1)\!-\!$simplex $x_{n+1}\in X_{n+1}$ such that $d^{n+1}_i(x_{n+1})=x_n^i$ for $0\leq j \leq n$. We denote $d_{n+1}^{n+1}(x_{n+1})$ by $h(x_n,x'_n)$, for such $x_{n+1}$.
\end{definition}
\tcr{This definition is not canonical, the missing position and order of the tuple can be swapped round}.\\
\noindent Notice that such $x_{n+1}$ is not necessary unique for given $x_n$ and $x'_n$. However, the below lemmas show that the $h(x_n,x'_n)\in X_n$ is a loop at $x_0$ for every such $x_{n+1}$, and that the equivalence class $[\iota_{h(x_n,x'_n)}]$(rel $\partial\Delta^n$) is independent of the choice of the representatives $x_n$ and $x'_n$ of the equivalence classes $[\iota_{x_n}]$ and $[\iota_{x'_n}]$ (rel $\partial\Delta^n$), it is also independent of the choice of such $x_{n+1}$ introduced in the above definition. Then, we show that this definition gives rise to a group structure on $\pi_n(X,x_0)$.
\begin{lemma}
Let $X$ be a simplicial set, $x_0\in X_0$, then $h(x_n,x'_n)$, defined above, is a loop at $x_0$.
\end{lemma}
\begin{proof}
Using lemma \ref{Lem:LoopBoundary}, it is enough to show that $\forall 0\leq i\leq n,\ d^n_i(h(x_n,x'_n))=\iota_{x_0,n-1}(\sigma_{n-1})$ in order to show that $h(x_n,x'_n)$ is a loop at $x_0$. $\forall 0\leq i\leq n$:
$$
d_i^n(h(x_n,x'_n))=d_i^n(d_{n+1}^{n+1}(x_{n+1}))=d_n^n(d_i^{n+1}(x_{n+1}))=d_n^n(x_n^i)=\iota_{x_0,n-1}(\sigma_{n-1}).
$$
Hence, $h(x_n,x'_n)$ goes though $x_0$.
\end{proof}
\begin{lemma}
Let $X$ be a simplicial set, $x_0\in X_0$, then $[\iota_{h(x_n,x'_n)}]$ (rel $\partial\Delta^n$) is independent of the choice of representative $x_n$ and $x'_n$ in the homotopy classes $[\iota_{x_n}]$ and $[\iota_{x'_n}]$. Moreover, it is independent of the choice of $x_{n+1}$ in definition \ref{Def:P_nOperation}.
\end{lemma}
\begin{proof}
\tcb{Jardine Lemma 7.1}
\end{proof}

\noindent \tcr{The group operation is not independent of the order, but about positioning}.
\begin{lemma}
Let $X$ be a Kan fibration, $x_0\in X_0$, then $\forall n\in \ZZ_{\geq 1}$, $\pi_n(X,x_0)$, have a \tcr{canonical} group structure given by: $\forall [\iota_{x_n}],[\iota_{x'_n}] \in \pi_n(X,x_0)$
$$
[\iota_{x_n}]\cdot[\iota_{x'_n}]=[h(x_n,x'_n)].
$$
Moreover, for $n\geq 0$, the resulting group is abelian.
\end{lemma}
\begin{proof}
\tcb{Jardine 7.2}
\end{proof}
\tcr{use the above lemma to get the needed non constant homotopy from $f$ to itself.}
\begin{lemma}\label{Lem:PiIsomorphism}
Let $X$ be a Kan fibration, $x_0\in X_0$, then $\forall n\in \ZZ_{\geq 0}$, there is a canonical bijection $\pi_n(X,x_0)\cong \pi_n(\mid X\mid ,x_0)$, which is a group isomorphism for $n\geq 1$.
\end{lemma}
\begin{proof}
\tcb{Joe 2.4.44, Hovey 3.6.3}
\end{proof}
 
\noindent \tcb{ calculate simplicial groups for $\ast$}\\

\begin{proposition}
Let $X$ be a non-empty Kan fibrant with no non-trivial simplicial homotopy groups. Then the unique morphism $X\rightarrow \ast$ is in $\II\!-\!$inj.
\end{proposition} 
\begin{proof}
\tcb{Hovey 3.4.7}
\end{proof}
 
\subsection{Minimal Fibrations}
\noindent Having the above isomorphism in mind, one might wonder if weak fibration coincide with Kan fibration that induces isomorphisms on the level of simplicial homotopy groups. However, the domain and codomain of Kan fibration are not necessary Kan fibrant. Hence, simplicial homotopy groups are not necessarily defined for its domain and codomain.
Therefore, minimal fibrations have been introduced to overcome this difficulty. We recall the definition and main properties below. Roughly speaking, the idea of minimal fibration is to distinguish Kan fibrations that induces an isomorphism fibre-wise on level of simplicial homotopy groups. Since, $\pi_n(\ast,\ast)$ is trivial, so it makes sense to define minimal fibrations to be Kan fibrations with trivial simplicial group fibre-wise. Minimal fibrations will also provide a sufficient criteria for morphisms to have the RLP with respect to $\II$, i.e. being in $\II\!-\!$inj.



\tcr{study based on the set of vertices of the fiber}
\begin{definition}
Let $q:X\rightarrow Y$ be a Kan fibration, we say that $q$ is a minimal fibration if every fibre-wise homotopic simplices of $X$ (rel $\partial\Delta^n$) with respect to $q$ coincide, i.e. $\forall x_n,x'_n\in X_n$  such that $x_n\sim_q x'_n$ (rel $\partial\Delta^n$), then one has $x_n=x'_n$.
\end{definition}
\begin{lemma}
Let $q:X\rightarrow Y$ be a fibration. Then, the following are equivalent:
\begin{enumerate}
\item $q$ is a minimal fibration.
\item For every fibre-wise homotopy $H\in \Hom(\Delta^n,X)_1$ (rel $\partial\Delta^n$) with respect to $q$, the below diagram commutes:
\[
\xymatrix{
\Delta^n\times\Delta^0\ar@<+2pt>[rr]^{id_L\times \Delta^{\partial_1^1}}
\ar@<-2pt>[rr]_{id_L\times \Delta^{\partial_1^0}}
&&\Delta^n\times\Delta^1\ar[rr]^H&&X.}
\]
\item ...Path components.
\end{enumerate}
\end{lemma}
\begin{proof}
\tcb{Joe 2.4.46}
\end{proof}
\begin{definition}
A Kan fibrant $X$ is called a minimal fibrant if the unique morphism $X\rightarrow \ast$ is a minimal fibration.
\end{definition}
\begin{cexample}
One can see easily that the fibrant $\Delta^n$ is not a minimal fibrant.
\end{cexample}
\noindent One can think of minimal fibration as simplicial equivalent to covering spaces, where there is no path between points of the fibres at a given point. However, in the case of minimal fibrations, the fibre can be empty.

\begin{lemma}\label{Lem:MiniClosed}
Minimal fibrations are closed under pullback, and retract.
\end{lemma}
\begin{proof}
\tcb{Joe 2.4.48}
\end{proof}

\begin{definition}[fibre-wise trivial]
Let $p:X\rightarrow Y$ be a fibration. We say that $p$ is fibre-wise trivial iff $\forall n\in \ZZ_{\geq 0}$, $y_n\in Y_n$, the pull back fibration $\Delta^n\times_Y X\rightarrow \Delta^n$ is isomorphic to a product fibration $\Delta^n\times F\stackrel{pr_1}{\rightarrow}\Delta^n$ over $\Delta^n$, i.e. there is an isomorphism of simplicial sets $\Delta^n\times F\rightarrow \Delta^n\times_Y X$ that makes the below diagram commute:
\[
\xymatrix{\Delta^n\times F\ar[rr]\ar[rd]_{pr_1}&&\Delta^n\times_Y X\ar[ld]\\
&\Delta^n&}
\]
%iff $\forall y_0\in Y_0$, $\forall x_0\in X_{y_0,0}$, $\pi_n(X_{y_0},x_0)$ is trivial.
\end{definition}
\tcr{explain, Hovey 3.5.1}



\begin{lemma}
Let $q:X\rightarrow Y$ be a minimal fibration. Then, $q$ is fibre-wise trivial.
\end{lemma}
\begin{proof}
\tcr{type}
\end{proof}





\tcr{Motivation}
\noindent One knows from topological homotopy theory that deformation retraction are homotopy of a topological space with respect to a subspace that is a retraction of the original space. This notion can be extended to simplicial sets in the below manner:
\begin{definition}[Strong Deformation Retract]
Let $i:K\rightarrow L$ be an inclusion of simplicial sets. $K$ is said to be a strong deformation retract of $L$, if there exist a retraction $r:L\rightarrow K$, i.e. $ri=id_K$, and a homotopy $H:L\times\Delta^1\rightarrow L$ relative to $K$ from $id_L$ to $ir$.
\end{definition}


\begin{definition}[Strong Fibre-wise Deformation Retract]
Let $p:X\rightarrow P$ be a Kan fibration. A morphism of simplicial sets $q:Z\rightarrow Y$ is  said to be a strong fibre-wise deformation retract of $p$, if there exist an inclusion of simplicial sets $i:Z\rightarrow X$, and retraction $r:X\rightarrow Z$ that makes $q$ a retract of $p$, and a homotopy $H:X\times\Delta^1\rightarrow X$ with respect to $p$ relative to $Z$ from $id_X$ to $ir$.
\end{definition}


\begin{proposition}
Let $p:X\rightarrow Y$ be a Kan fibration. Then, $p$ has a strong fibre-wise deformation retract $q:Z\rightarrow Y$ which is a minimal fibration.
\end{proposition}
\begin{proof}
\tcb{Jardine 10.}
\end{proof}


\begin{lemma}
Let $p:X\rightarrow Y$ be a Kan fibration, $x_n,x'_n\in X_n$ degenerate such that $x_n\sim_p x'_n$. Then, $x_n=x'_n$.
\end{lemma}
\begin{proof}
\tcb{Hovey 3.5.8}
\end{proof}

i.e. minimal fibration imposes its condition only on non-degenerate simplices.

\begin{theorem}
Let $p:X\rightarrow Y$ be a Kan fibration. Then, $p$ factorise as $\xymatrix{X\ar[r]^{r}&X'\ar[r]^{p'}&Y}$, with $p'$ being a minimal fibration, and $r$ a retraction of the simplicial sub-set $i:X'\hookrightarrow X$ such that $r\in\II\!-\!$inj.
\end{theorem}
\begin{proof}
\tcb{Hovey 3.5.9}
\end{proof}

\noindent The below corollary provides a sufficient criteria for fibrations to be in $\II-\!\!$inj
\begin{corollary}\label{Cor:IinjCriteria}
Let $p:X\rightarrow Y$ be a Kan fibration having a non-empty fibres with no non-trivial simplicial homotopy groups $\forall y_0\in Y_0$ . Then, $p\in \II-\!\!$inj.
\end{corollary}
\begin{proof}
\tcb{Hovey 3.5.10}
\end{proof}

\begin{theorem}[Gabriel-Zisman]
Let $q:X\rightarrow Y$ be a minimal fibration. Then, the realisation $\mid q\mid:\mid X\mid\rightarrow \mid Y\mid$ is a Serre fibration.
\end{theorem}
\begin{proof}
\tcb{Jardine 10.9}
\end{proof}

\begin{corollary}[Quillen's Theorem]\label{Th:Quillen}
Let $p:X\rightarrow Y$ be a morphism of simplicial sets. Then, $p$ is a Kan fibration iff $\mid f\mid:\mid X\mid\rightarrow\mid Y\mid$ is a Serre fibration.
\end{corollary}
\begin{proof}
\tcb{Jardine 10.10+..Hovey 3.6.2}
\end{proof}

then lemma before becomes a corrolry is a result of the fact that all topological spaces are fibrant.

\begin{corollary}\label{SSets:WeakFib:Iinj}
Let $p:X\rightarrow Y$ be a morphism of simplicial sets. Then, $p$ is a weak fibration iff it is in $\II\!-\!$inj.
\end{corollary}
\begin{proof}[following Joe 2.4.60/2.4.7, Hovey 3.6.4]
Suppose that $p$ is a weak fibration. Using corollary \ref{Cor:IinjCriteria}, it is sufficient to show that fibres of $p$ are non-empty and have no non-trivial simplicial homotopy groups. $\forall y_0\in Y_0$ , consider the pullback
\[
\xymatrix{
X_{y_0}\ar[r]\ar[d]&X\ar[d]^p\\
\Delta^0\ar[r]_{\iota_{y_0}}&Y}
\]
Then $X_{y_0}$ is a fibrant, by \ref{Lem:InjProjCellClosed}\ref{itm:InjPullProjPush}. Since, the geometrical realisation functor preserves pullbacks, lemma \ref{}, we have the following pullback square
\[
\xymatrix{
\mid X_{y_0}\mid \ar[r]\ar[d]&\mid X\mid \ar[d]^{\mid p\mid }\\
\mid \Delta^0\mid \ar[r]_{\mid \iota_{y_0}\mid }&\mid Y\mid 
}
\]
then, by corollary \ref{Th:Quillen}, $\mid p \mid$ is a Serre fibration. $\mid X_{y_0}\mid $ is the fibre of $\mid p\mid $ over $\mid \iota_{y_0}\mid$. \tcb{Since $\mid p \mid$ is a weak equivalence,} then \tcr{weak} Serre fibrant $\mid X_{y_0}\mid $ is \tcb{non-empty} and has no non-trivial homotopy groups. Then, $X_{y_0}$ is non-empty, and by lemma \ref{Lem:PiIsomorphism} it has no non-trivial simplicial homotopy group. Hence, by corollary \ref{Cor:IinjCriteria}, we see that $p\in \II\!-\!$inj.\\

%%%%%%%%%%%%%%%%%%%Let $p\in \II\!-\!$inj.
%%%%%%%%%%%\tcb{type}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%\tcr{...Joe 2.4.60/2.4.7, Hovey 3.6.4}
\end{proof}

\noindent Since every fibration $p$ factorises as $p=q r$  where $r\in \II\!-\!$inj and $q$ a minimal fibration, and $\II\!-\!$inj are weak fibrations. Then, the information id a fibration is a weak fibration is encoded in its minimal fibration factor.


%%%%%%%\subsection{Whitehead Theorem of Simplicial Sets}

\section{The Canonical Model Structure}
\begin{theorem}[Organising Theorem]\label{Th_SimpSets_ModelStructure}
The category of simplicial sets $\SSets$, together with the classes of Kan fibrations, cofibrations, and weak equivalences, studied above, forms a model category.
\end{theorem}
\begin{proof}
We will use theorem \ref{CofGen:Creteria}.
We examine below that the five axioms of model category, Def. \ref{Def_CM}, are satisfied:
\begin{enumerate}[label=\textup{\textbf{CM\arabic*}},ref=\textup{\textbf{CM\arabic*}}]
\item\label{SSets:CM1}  Since the category of small sets $\Sets$ is complete and cocomplete, then by lemma \ref{Lem_FuncCat_complete}, $\SSets$ is complete and cocomplete. Furthermore, for any small diagram $F:J\rightarrow \SSets$, we have
\[
\left(\displaystyle\lim_J F \right)_n=\displaystyle\lim_J F_n \ \text{, and }\ \left(\displaystyle\colim_J F \right)_n=\displaystyle\colim_J F_n 
\]
for every $n\in\ZZ_{\geq 0}$, and the axiom \ref{itm:CM1} holds.
\item\label{SSets:CM2}(2 out of 3) For any commutative diagram of simplicial sets
\[
\xymatrix @R=.25pc
{X\ar[rr]^{gf}\ar[rd]_{f}&&Z\\
&Y\ar[ur]_g&
}.
\]
Using the realisation functor, we get the commutative diagram of topological spaces
\[
\xymatrix @R=.25pc
{\mid X\mid \ar[rr]^{\mid g\mid\mid f\mid}\ar[rd]_{\mid f\mid}&&\mid Z\mid\\
&\mid Y\mid\ar[ur]_{\mid g\mid}&
}.
\]
If any two of $f,g$ or $g f$ are weak equivalences of simplicial sets, then by definition two maps of $\mid f\mid,\mid g\mid$ and $\mid g\mid \mid f\mid$ are weak equivalences of topological spaces. Using the model structure on $\Top$, the third map is a weak equivalence of topological spaces. Hence, the third corresponding morphism is a weak equivalence of simplicial sets, and the axiom \ref{itm:CM2} holds.
\item\label{SSets:CM3}(Stability under retract) Let $f:X\rightarrow Y,$ and $g:X'\rightarrow Y'$ be morphisms of simplicial sets such that $f$ is a retract of $g$. Then, we have the commutative diagram:
\[
\xymatrix{
X\ar[r]^{d_0}\ar[d]_f&X'\ar[r]^{r_0}\ar[d]|g&X\ar[d]^f\\
Y\ar[r]_{d_1}\ar@{}[ru]|{D}&Y'\ar[r]_{r_1}\ar@{}[ru]|{R}&Y}
\]
with the horizontal composite being the identities.
Then:
	\begin{enumerate}
	\item\label{SSets:CM3.1} Suppose that $g$ is a cofibration, then $g\in \II\!-\!$cof, then by lemma \ref{InjProjRetractStatbility} $f$, a retract of $g$, is in $\II\!-\!$cof, i.e. $f$ is a cofibration.
	\item\label{SSets:CM3.2} Suppose that $g$ is a Kan fibration, then $g\in \JJ\!-\!$inj, then by lemma \ref{InjProjRetractStatbility} $f$, a retract of $g$ is in $\JJ\!-\!$inj, i.e. $f$ is a fibration.\\
Notice that, here, the cofibration and fibration is quite formal, does not depend on nature of morphisms of simplicial sets, but rather on being defined as the morphisms with LLP or RLP with respect to some classes of morphisms.
	\item\label{SSets:CM3.3} Suppose that $g$ is a weak equivalence, applying the realisation, we have the commutative diagram in $\Top$, with the horizontal composite being the identities:
\[
\xymatrix{
\mid X\mid\ar[r]^{\mid d_0\mid}\ar[d]_{\mid f\mid}&\mid X'\mid\ar[r]^{\mid r_0\mid}\ar[d]|{\mid d_0\mid}&\mid X\mid\ar[d]^{\mid f\mid}\\
\mid Y\mid\ar[r]_{\mid d_1\mid}\ar@{}[ru]|{\mid D\mid}&\mid Y'\mid\ar[r]_{\mid r_1\mid}\ar@{}[ru]|{\mid R \mid}&Y
}
\]	
Since, $g$ is a weak equivalence of simplicial sets, then $\mid g\mid$ is a a weak equivalence of topological spaces. Since weak equivalence of topological spaces is closed under retract, then so is $\mid f\mid$. Hence, $f$ is a weak equivalence of simplicial sets, and \ref{itm:CM3} holds.
% Then, we need to show that $\mid f\mid$'s are bijections $\forall n\in\ZZ_{\geq 0}$. $\forall n\in\ZZ_{\geq 0}$, we can use the fact the horizontal composite being the identities to show that $\mid r_1 \mid$'s are surjections, and $\mid d_1\mid$'s are injections to show that $\mid f\mid$ is surjective, injective, respectively. Or rather, we show that $\mid f\mid$ has the inverse $\mid r_0\mid\mid g\mid^{-1}\mid d_0\mid$, that:
%\[\mid g\mid \mid d_0\mid=\mid d_1\mid \mid f\mid\text{, and }\mid f\mid \mid r_0\mid=\mid r_1\mid \mid g\mid\]
%i.e.
%\[  \mid d_0\mid=\mid g\mid^{-1}\mid d_1\mid \mid f\mid\text{, and }\mid f\mid \mid r_0\mid \mid g\mid^{-1}=\mid r_1\mid\]
%Then,
%\[\mid f\mid \left(\mid r_0\mid\mid g\mid^{-1}\mid d_1\mid \right)=\mid r_1\mid \mid d_1\mid=id_{\mid Y\mid}.\]
%And, 
%\[\left(\mid r_0\mid\mid g\mid^{-1}\mid d_1\mid \right)   \mid f\mid= \mid r_0\mid\mid d_0\mid=id_{\mid X\mid}.\]
%Hence, $\mid f\mid$ is bijection for $n\in \ZZ_{\geq 0}$, and $f$ is a weak equivalence.
	\end{enumerate}
\item\label{SSets:CM4}(Lifting) Let $E$ be a commutative diagram of morphisms of simplicial sets
\[\xymatrix{
A\ar[r]^{e_0}\ar[d]_i&X\ar[d]^p\\
B\ar[r]_{e_1}\ar@{}[ru]|E&Y
}
\]
Where $i$ is a cofibration, $p$ is a fibration, and either of $i$ or $p$ is a weak equivalence. Then we need to show that $E$ admits a lift $h:B\rightarrow X$.\\

On the one hand, if $p$ is a weak equivalence then $p\in \II\!-\!$inj, whereas $i\in \II\!-\!$cof$=(\II\!-\!$inj$)\!-\!$proj. Hence, $E$ admits a lift.

On the other hand, if $i$ is a weak equivalence. Then, by \ref{SSets:CM5} proved above, $i$ factorise as $i=q j$ where $q$ is a fibration and $j$ is an anodyne extension, hence a weak cofibration. Then, by the (2 out of 3)  property proved above $q$ weak fibration. Then, using the above argument for the diagram 
\[\xymatrix{
A\ar[r]^{j}\ar[d]_i&A'\ar[d]^q\\
B\ar[r]_{id_B}\ar@{}[ru]|E&B
}
\]
we have a lift $h':B\rightarrow A'$. Then, $i$ is a retract of $j$, so $i$ is an anodyne extension. Hence, it has Since anodyne extensions have the LLP with respect to all fibrations, hence $E$ admits a lift.
\item\label{SSets:CM5}(factorisation) Let $f:X\rightarrow Y$ be a morphism of simplicial sets. Since every simplicial sets is small, then in particular the domains of $\II$ are small-relative to $\II\!-\!$cell. Then, by the small object argument, $f$ factorise as:
\[
f=p i, p\in \II\!-\!\text{inj}, i\in \II\!-\!\text{cell}.
\]
Since $\II\!-\!$cell$\subset\II\!-\!$cof. Then, $i$ is a cofibration. Also, $\II\!-\!$inj are weak fibrations, hence $p$ is weak fibration.\\
The domains of $\JJ$ are small-relative to $\JJ\!-\!$cell, hence $f$ also factorise as:
\[
f=q j, q\in \JJ\!-\!\text{inj}, j\in \JJ\!-\!\text{cell}.
\]
$\JJ\!-\!$inj are Kan fibrations, hence $q$ is a fibration. Also, $\JJ\!-\!$cell$\subset\JJ\!-\!$cof, i.e. $j$ is anodyne extension, hence a weak cofibration.

\end{enumerate}
\end{proof}
\subsection{Cofibrations}
As seen above the proof of the factorisation axiom \ref{SSets:CM5} is based on using transfinite small object argument on the classes of morphisms $\II$ and $\JJ$. Therefore, we recall hereby that $\II$ and $\JJ$ admit transfinite small object argument in the below lemma.
\Lem{}{
Let $\kappa$ be an infinite cardinal. Then, $\kappa\!-\!$bounded simplicial sets are small relative to cofibrations.
}{SSet:Bounded:SmallRelative}
\begin{proof}
Let $A$ be a $\kappa\!-\!$bounded simplicial set. We need to show that for any $\kappa\!-\!$ filtered ordinal $\lambda$, and any $\lambda\!-\!$sequence $F:\lambda\rightarrow \SSets$ of cofibrations, the canonical morphism \eqref{eq:Small:Relative}
\[
\vartheta: \displaystyle\colim_\lambda \SSets(A,F_-)\rightarrow
\SSets(A,\displaystyle\colim_\lambda F)
\]
is a bijection. Since colimits of sets are quotients of disjoint unions, then to show that $\vartheta$ is surjective, we need to show that for any morphism of simplicial set $f:A\to \displaystyle\colim_\lambda F$ there exist $\beta<\lambda$ and morphism of simplicial set $A \to F_\beta$, for which we have the commutative diagram
\[
\xymatrix{
A\ar[dr]\ar@/^/[rrrd]^f\\
&F_\beta\ar[rr]|(.4){\ \ldots\ }&&\displaystyle\colim_\lambda F
}
\]
where $F_\beta\to \displaystyle\colim_\lambda F$ is the evident transfinite composition.\\

Since colimits in $\SSets$ are given object-wise, then $\forall n\in \ZZ_{\geq 0}$, $f$ induces maps of sets
\[
f_{n}:A_n\to \displaystyle\colim_\lambda (F_-)_n
\]
where $(F_-)_n:\lambda\to \Sets$ is the evident $\lambda\!-\!$sequences of injections of sets. The transfinite composition of cofibrations is so as we see later in lemma \ref{Lem:SSets:Cof:Closed}, also the same holds for transfinite composition of injections of sets as in lemma \ref{Lem:Cat:Mon:Closed}. Hence, all evident transfinite compositions 
\[
 F_\beta\hookrightarrow \colim_{\beta'} F \text{ , and } 
(F_\beta)_n\hookrightarrow\colim_{\beta'}(F_-)_n
\]
are cofibrations and injections, respectively, $\forall \beta<\beta'\leq\lambda$. Then, by the definition of colimits of sets $\forall x_{n}\in \displaystyle\colim_\lambda (F_-)_n$ there exist $\beta_{x_{n}}<\lambda$ such that $x_{n}\in (F_{\beta_{{n}}})_n    \subseteq \displaystyle\colim_\lambda (F_-)_n$. In particular, $\forall a_{n}\in A_n$, there exists $\beta_{a_{n}}<\lambda$ such that $f_{n}(a_{n})\in (F_{\beta_{a_{n}}})_n$. Then, setting 
\[
\beta_{n}=\sup\{\beta_{a_{n}}|a_{n}\in A_n \}
\]
we notice that $\beta_{n}<\lambda$ because $\lambda$ is a $\kappa\!-\!$filtered ordinal, $\beta_{a_{n}}<\lambda$ and $\mid A_n \mid <\kappa$, recall definition \ref{FilteredOrdinals}. Then, $f_{n}(a_{n})\in (F_{\beta_{a_{n}}})_n\subset (F_{\beta_{{n}}})_n$. Hence, maps $f_{n}$ factorise through $(F_{\beta_{{n}}})_n$. Now, put
\[
\beta=\sup\{\beta_{n}|n\in \ZZ_{\geq 0}\}.
\]
Since $\mid \ZZ_{\geq 0}\mid= \aleph_0\leq \kappa$, and $\beta_{n}<\lambda, \forall n\in \ZZ_{\geq 0}$, then $\beta<\lambda$. Hence, all maps $f_{n}$ factorise through $g_{n}:A\to(F_\beta)_n$. Now, in order to show that $f$ itself factorises through $F_\beta$, it is enough to show that $g_{n}$'s give rise to a morphism of simplicial set, i.e. they should be natural in their argument, which is evident from the commutativity of the below diagram, and the fact that $(F_\beta)_m\hookrightarrow \displaystyle\colim_\lambda (F_-)_m$ is an injection of sets (i.e. a monomorphism, cancels from the left), for any $\mu:[m]\to[n]$ in $\Simp$
\[
\xymatrix{
A_n\ar[dr]_{g_{n}}\ar@/^/[rrrd]^(.6){f_{n}}\ar[dd]_{A_\mu}\\
&(F_\beta)_n\ar[dd]^(.4){(F_\beta)_\mu}\ar@{^(->}[rr]|(.4){\ \ldots\ }&&\displaystyle\colim_\lambda (F_-)_n\ar[dd]^{\displaystyle\colim_\lambda (F_-)_\mu}\\
A_m\ar[dr]_{g_{m}}\ar@/^/[rrrd]|(.3575)\hole|(.6){f_{m}}\\
&(F_\beta)_m\ar@{^(->}[rr]|(.4){\ \ldots\ }&&\displaystyle\colim_\lambda (F_-)_m
}
.
\]
Therefore, there exists $g:A\to F_\beta$, given object-wise above, such that $f=\vartheta([g])$, where $[g]$ is the class of $g$ in $\displaystyle\colim_\lambda \SSets(A,F_-)$.

On the other hand, to show that $\vartheta$ is injective, let $g:A\to F_\beta,g':A\to F_{\beta'}$ such that $\vartheta([g])=\vartheta([g])$, then we need to show that $[g]=[g']$. By the definition of $\vartheta$
\[
c_\beta^\lambda \circ g=\vartheta(g)=\vartheta(g')=c_{\beta'}^\lambda \circ g'
\]
where $c_\beta^\lambda:F_\beta\hookrightarrow \displaystyle\colim_\lambda F$, $c_{\beta'}^\lambda:F_{\beta'}\hookrightarrow \displaystyle\colim_\lambda F$ are the evident transfinite compositions. Without loose of generality, let $\beta\leq \beta'$, since $c_{\beta}^\lambda=c_{\beta'}^\lambda \circ c_{\beta}^{\beta'}$, and $c_{\beta'}^\lambda$ cancels from the left for being a cofibration (a monomorphism in $\SSets$), then we have $c_{\beta}^{\beta'}\circ g=g'$. But, $[c_{\beta}^{\beta'}\circ g]=[g]$, hence $[g]=[g']$.
\end{proof}
Notice that the converse of the above statement is not true as shown in the below example.
\CEx{}{\tcr{????}}{}


\Cor{}{The classes $\II$ and $\JJ$ defined above admits the transfinite small object argument in $\SPre(\bcC)$.}{SSets:IJ:SmallObjArg}
\begin{proof}
Direct result of lemma \ref{Lem:SSet:Bounded:SmallRelative}, above. That, $\II$ and $\JJ$ are classes of cofibrations, and for $\kappa$ infinite, $\mid\partial\Delta^n\mid<\kappa$, and $\mid \Lambda^n_r\mid <\kappa$, for every $n\in \ZZ_{\geq 0}, 0\leq r\leq n$.
\end{proof}


%http://ncatlab.org/nlab/show/model+structure+on+simplicial+sets
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\tcr{show that $L\times \Delta^1$ is a cylinder object, and $\Hom(\Delta^1,X)$ is a path object in $\SSets$, and that simplicial homotopy is a left homotopy in $\SSets$. Moreover, having $\SSets$ closed Cartesian, enable use the adjunction from $-\times \Delta^1$ to $\Hom(\Delta^1,-)$ in order to interchange between right and left homotopy....}


\section{Kan Extension $Ex^\infty$}

Kan developed a method of calculating the fibrant replacements in Quillen's model category of simplicial sets. It is given by Kan ${Ex}^\infty$ functor \tcr{(cite)}. For a simplicial set $X$, ${Ex}^\infty(X)$ is the colimit of an injective system
\[
\xymatrix{
X\ar[r]^{j_X}&Ex (X)\ar[r]^{j_{Ex (X)}}&Ex^2 (X)\ar[r]^{j_{Ex^2 (X)}}&Ex^3 (X)\ar[r]^{j_{Ex^3 (X)}}&\ldots
}
\]
of inclusions, where $Ex$ is defied using the subdivision functor $sd$, by
\[Ex(X)_n =\SSets(\Delta^n, Ex(X))= \SSets(sd \Delta^n, X).\]
with faces and degenerations given by pre-composition with cofaces and codegenerations of $sd \Delta^n$'s.
Recall that $ sd \Delta^n = \bcN nd \Delta^n$, where $\bcN$ is the nerve functor, and the $nd-$ is the poset of nondegenerate simplices. 

It is worth noticing that Kan extension of Kan fibrants is a fibrant that is more involved and does not coincide with the original fibrant, even for very 'simple' simplicial sets, as seen in the below example.
\Ex{}{
$\Delta^1$ is a Kan fibrant. Hereby we just calculate $sk_1(Ex(\Delta^1))$, and using the injectivity $j$'s above, deduce  that $\Delta^1 \neq Ex^\infty(\Delta^1)$. For $n=0$
\[nd \Delta^0= \xymatrix{\stackrel{id_{[0]}}{\bullet} \ar@(dr,dl)}\]\vspace{.5cm}
Therefore, $sd \Delta^0=\bcN nd \Delta^0=\ast$, hence $Ex(\Delta^1)_0=\SSets(\ast,\Delta^1)=\Delta^1_0=\{\partial_1^0,\partial_1^1\}$. Then, for $n=1$
\[nd \Delta^1= \xymatrix{ \stackrel{\partial_1^0}{\bullet} \ar@(dr,dl)\ar[r]&\stackrel{id_{[1]}}{\bullet} \ar@(dr,dl)&\stackrel{\partial_1^1}{\bullet} \ar@(dr,dl)\ar[l]}.
\]
Then,
\[
\begin{array}{ll}
(sd \Delta^1)_0=&\{\partial_1^0,id_{[1]},\partial_1^1\}.\\
(sd \Delta^1)_1=&\{id_{\partial_1^0},(\partial_1^0,id_{[1]}),   id_{id_{[1]}} ,(\partial_1^1,id_{[1]}) , id_{\partial_1^1}\}.\\
(sd \Delta^1)_2=&\left\lbrace(id_{\partial_1^0},id_{\partial_1^0}), (id_{\partial_1^0},(\partial_1^0,id_{[1]})),((\partial_1^0,id_{[1]}),id_{id_{[1]}}),(id_{id_{[1]}},id_{id_{[1]}}),((\partial_1^1,id_{[1]}),id_{id_{[1]}}),\right. \\&\left.(id_{\partial_1^1},(\partial_1^1,id_{[1]})),(id_{\partial_1^1},id_{\partial_1^1})) \right\rbrace. \\
\ \ \ \vdots&
\end{array}
\]
To understand a simplicial set is basically equivalent to understanding its faces and degenerations. Hence, the faces and degenerations of $sd \Delta^1$ are highlighted below, up to $2\!-\!$simplicies.\\
$s^0_0$ maps the $0\!-\!$simplicies to their identity maps, i.e. it maps $\partial_1^0,id_{[1]},\partial_1^1$ to $id_{\partial_1^0},id_{id_{[1]}},id_{\partial_1^1}$, respectively. Hence, there are only two non-degenerate $1\!-\!$simplicies in $sd \Delta^1$, namely $(\partial_1^0,id_{[1]}),(\partial_1^1,id_{[1]})$.\\
$d^1_0,d^1_1$ maps $1\!-\!$simplicies to their codomain, domain respectively.\\
$s_0^1,s^1_1$ maps $1\!-\!$simplicies to the $1\!-\!$simplicies obtained by insetting identity morphisms at the domain, codomain, respectively. In particular,
$s_0^1$ maps 
\[
\begin{array}{c}
\underbrace{id_{\partial_1^0},(\partial_1^0,id_{[1]}),   id_{id_{[1]}} ,(\partial_1^1,id_{[1]}) , id_{\partial_1^1}}\\
\downmapsto\\
(id_{\partial_1^0},id_{\partial_1^0}),(id_{\partial_1^0},(\partial_1^0,id_{[1]})),   (id_{id_{[1]}},id_{id_{[1]}}) ,(id_{\partial_1^1},(\partial_1^1,id_{[1]})) , (id_{\partial_1^1},id_{\partial_1^1})
\end{array}
\]
and $s^1_1$ maps 
\[
\begin{array}{c}
\underbrace{id_{\partial_1^0},(\partial_1^0,id_{[1]}),   id_{id_{[1]}} ,(\partial_1^1,id_{[1]}) , id_{\partial_1^1}}\\
\downmapsto\\
(id_{\partial_1^0},id_{\partial_1^0}),((\partial_1^0,id_{[1]}),id_{id_{[1]}}) ,   (id_{id_{[1]}},id_{id_{[1]}})  ,((\partial_1^1,id_{[1]}),id_{id_{[1]}}) , (id_{\partial_1^1},id_{\partial_1^1}).
\end{array}
\]
Then, all $2\!-\!$simplicies of $sd \Delta^1$ are degenerate. One can also see that all $n\!-\!$simplicies of $sd \Delta^1$ are degenerate for $n\geq 2$. 
$d_0^2,d_2^2$ maps $2\!-\!$simplicies to their codomain, domain respectively, whereas $d_1^2$ maps $2\!-\!$simplicies to their composition. Hence $sd \Delta^1$ has only two non-degenerate simplicies, in addition to its three vertices with
\[d^1_0((\partial_1^0,id_{[1]}))=id_{[1]}, d^1_0((\partial_1^1,id_{[1]}))=id_{[1]} \ \text{ , and } \ 
d_1^1((\partial_1^0,id_{[1]}))=\partial_1^0, d_1^1((\partial_1^1,id_{[1]}))=\partial_1^1
\]
Hence, it has the geometric realisation
\[ \xymatrix{ \stackrel{\partial_1^0}{\bullet} \ar@{-}[r]_{(\partial_1^0,id_{[1]})}&\stackrel{id_{[1]}}{\bullet} &\stackrel{\partial_1^1}{\bullet} \ar@{-}[l]^{(\partial_1^1,id_{[1]})}}.
\]
Since $\im d_0^1 \bigcup \im d_1^1= (sd \Delta^1)_0$, then morphisms of simplicial sets from $sd \Delta^1$ are determined by the image of the two $1\!-\!$simplices $(\partial_1^0,id_{[1]})$ and $(\partial_1^1,id_{[1]})$. $\Delta^1$ has three $1\!-\!$simplices
\[
\Delta^1_1=\{\partial_1^0\sigma_0^0, id_{[1]}, \partial_1^1\sigma_0^0\}
\]
Then a straightforward verification shows that there are exactly five morphisms of simplicial sets from $sd\Delta^1$ to $\Delta^1$, given below by a visualisation of their images
\begin{center}
\begin{tabular}{ c | c | c | c | c }
A&B&C&D&E\\
\hline
$\xymatrix@R=.5pc@C=.75 pc{\ &&\\
\partial_0^1\ar@{-}[r]&id_{[1]}&\ar@{-}[l] \partial_1^1
}$&
$\xymatrix@R=.5pc@C=.75 pc{&id_{[1]}&\\
\partial_0^1\ar@{-}[ru]&&\ar@{-}[ul] \partial_1^1
}$&
$\xymatrix@R=.5pc@C=.75 pc{\partial_0^1\ar@{-}[r]&id_{[1]}&\\
&&\ar@{-}[ul] \partial_1^1
}$& 
$\xymatrix@R=.5pc@C=.75 pc{&id_{[1]}&\ar@{-}[l] \partial_1^1\\
\partial_0^1\ar@{-}[ru]&&
}$& 
$\xymatrix@R=.5pc@C=.75 pc{\partial_0^1\ar@{-}[r]&id_{[1]}&\ar@{-}[l] \partial_1^1\\
&&
}$
\end{tabular}
\end{center}
where upper positions represent being mapped to $\partial_1^0\sigma_0^0$, diagonals to $id_[1]$, and lower positions to $\partial_1^1\sigma_0^0$. Since faces and degenerations of $Ex(\Delta^1)$ is given by pre-composition with cofaces and codegenerations of $sd \Delta^n$'s, then $sk^1(Ex(\Delta^1))$ has the geometric realisation visualised below:
\[
\xymatrix{\stackrel{\partial_1^0}{\bullet}\ar@{-}@/_/[rr]_D\ar@{-}@/^/[rr]^C&&\stackrel{\partial_1^0}{\bullet}\ar@{-}@(ur,dr)^B
}
\]
Then $\Delta^1\subsetneqq sk_1( Ex(\Delta^1))\subset Ex(\Delta^1)$. Since $Ex^\infty(\Delta^1)$ is the colimit if injective system of inclusions then, $\Delta^1 \subsetneqq Ex^\infty(\Delta^1)$.
}{}
\Rem{}{When calculating Kan extension for standard simplexes $\Delta^n$. Calculation can be simplified due to the fact that the nerve functor $\bcN$ is faithfully full and that $\bcN [n]=\Delta^n$. Then, 
\[
Ex(\Delta^n)_n=\Sets(sd \Delta^1,\Delta^1)=\Sets(\bcN nd\Delta^1,\bcN [n])\cong\Pos(nd\Delta^1,[n])
\]
as has been used in example \ref{Ex:KanExDelta1}.
}{}